160 THEORY OF HEAT. [CHAP. III. 



197. It is easy to see either by means of this equation, or 

 from Art. 171, that heat is propagated in this solid, by sepa 

 rating more and more from the origin, at the same time that it 

 is directed towards the infinite faces B and G. Each section 

 parallel to that of the base is traversed by a wave of heat which 

 is renewed at each instant with the same intensity: the intensity 

 diminishes as the section becomes more distant from the origin. 

 Similar movements are effected with respect to any plane parallel 

 to the infinite faces; each of these planes .is traversed by a con 

 stant wave which conveys its heat to the lateral masses. 



The developments contained in the preceding articles would 

 be unnecessary, if we had not to explain an entirely new theory, 

 whose principles it is requisite to fix. With that view we add 

 the following remarks. 



198. Each of the terms of equation (a) corresponds to only 

 one particular system of temperatures, which might exist in a 

 rectangular plate heated at its end, and whose infinite edges are 

 maintained at a constant temperature. Thus the equation 

 v = e~ x cos y represents the permanent temperatures, when the 

 points of the base A are subject to a fixed temperature, denoted 

 by cos y. We may now imagine the heated plate to be part of a 

 plane which is prolonged to infinity in all directions, and denoting 

 the co-ordinates of any point of this plane by x and y, and the 

 temperature of the same point by v t we may apply to the entire 

 plane the equation v = e~ x cos y ; by this means, the edges B and 

 G receive the constant temperature ; but it is not the same 

 with contiguous parts BB and CO ; they receive and keep lower 

 temperatures. The base A has at every point the permanent 

 temperature denoted by cos y, and the contiguous parts A A have 

 higher temperatures. If we construct the curved surface whose 

 vertical ordinate is equal to the permanent temperature at each 

 point of the plane, and if it be cut by a vertical plane passing 

 through the line A or parallel to that line, the form of the section 

 will be that of a trigonometrical line whose ordinate represents 

 the infinite and periodic series of cosines. If the same curved 

 surface be cut by a vertical plane parallel to the axis of x, the 

 form of the section will through its whole length be that of a 

 logarithmic curve. 



